Unifying Game Theoretic and Inclusive Fitness Models of Altruism

Evolutionary
Game Theory
ISCI 330
Lecture 17
1 ISCI 330 Lecture 17
Outline
• A bit about historical origins of
Evolutionary Game Theory
• Main (competing) theories about how
cooperation evolves
• iPD and other social dilemma games
• Evolutionary Stable Strategy (ESS)
• N-player PD (and other games)
• Simpson’s paradox and the
role of assortment
2 ISCI 330 Lecture 17
Evolution by Natural Selection
• Lewontin’s principles (from Darwin)
– 1) Phenotypic variation
– 2) Differential fitness
– 3) Heritability
• In Evolutionary Game Theory
– 1) Population of strategies
– 2) Utility determines number of offspring (fitness)
– 3) Strategies breed true
• Frequency-dependent selection
– One of the first examples is Fisher’s
sex ratio findings
– Introduces idea of strategic phenotypes
3 ISCI 330 Lecture 17
Ritualized Fighting
opponent’s behaviour
Dove
Dove
actor’s
behaviour
Hawk
Hawk
V/2
5
0
V
V/2-c
10
-5
• V = 10; c = 10
• The rare strategy has an advantage
(i.e. frequency dependent selection)
• Hawk-Dove, Chicken, Snowdrift, Brinkmanship
• If c < V / 2, then game is PD instead
4 ISCI 330 Lecture 17
George Price’s Contributions
• Evolutionary Game Theory
– Concept of an Evolutionary Stable Strategy
(ESS)
• Formal description of Natural Selection
and Fisher’s Fundamental Theorem
• Decomposition of selection at different
hierarchical levels using covariance
– Used to formalize Multilevel
Selection Theory
5 ISCI 330 Lecture 17
What special circumstances or
mechanisms thus favor cooperation?
Currently, evolutionary biology offers a set
of disparate explanations, and a general
framework for this breadth of models has
not emerged.
– Sachs et al. 2004, The Evolution of
Cooperation. QRB 79:135-160
6 ISCI 330 Lecture 17
Main Theories:
Evolution of Altruism
• Multilevel Selection
∆Q = ∆QB + ∆QW (Price Equation)
• Inclusive Fitness/Kin Selection
– wincl. = wdirect + windirect
∆Q > 0 if rb > c (Hamilton’s rule)
• Reciprocal Altruism
∆Q > 0 if altruists are sufficiently
compensated for their sacrifices
via reciprocity (ESS)
7 ISCI 330 Lecture 17
Prisoner’s Dilemma (PD)
Actor's Fitness (Utility)
opponent’s behavior
C
D
C
4
0
D
5
1
actor’s
behavior
8 ISCI 330 Lecture 17
Additive Prisoner’s Dilemma (PD)
Actor's Fitness (Utility)
opponent’s behaviour
C
D
contributes b contributes 0
actor’s
behaviour
C
sacrifices c
D
sacrifices 0
w0 + b – c
w0 – c
w0 + b
w0
4
• w0 = 1; b = 4; c = 1
9 ISCI 330 Lecture 17
5
0
1
Non-Additive PD
Actor's Fitness (Utility)
opponent’s behavior
C
D
contributes b contributes 0
actor’s
behavior
C
sacrifices c
D
sacrifices 0
w0 + b – c
(+d) 3
w0 + b
5
• w0 = 1; b = 4; c = 1; d = -1
10 ISCI 330 Lecture 17
w0 – c
0
w0
1
Main Theories:
Evolution of Altruism
• Multilevel Selection
– Predominate models are in terms of public good
• Inclusive Fitness/Kin Selection
– Predominate models is in terms of individual
contributions (b and c)
• Reciprocal Altruism
– Predominate models in terms of
iterated PD
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Evolutionarily Social Dilemma Games
• What features do Hawk-Dove and the PD
have in common?
– Cs do better in CC pairs than Ds do in DD pairs
– Ds do better than Cs in mixed pairs
• Given 4 utility levels (1st, 2nd, 3rd, 4th) how
many 2-player, symmetric games are there
that capture this idea of “social dilemma”?
• With a partner, find these
other games. Can you name
them?
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6 evolutionarily interesting
“social dilemmas”
• How do these games compare in terms
of
– Nash equilibria?
– Pareto optimality?
– Is it better to be rare or common?
• Consider populations of strategies
rather than 2-players
• Relative vs. Absolute fitness
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Common EGT Assumptions
• Population of strategies
• Replicator equations often assume
– infinite populations
– continuous (or discrete) time
– complete mixing (random interactions)
– strategies breed true (no sex)
14 ISCI 330 Lecture 17

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